Regular polygon conjecture for the first Dirac eigenvalue

Let $n\geq1$, let $P_n$ be an $n$-sided polygon, and let $P_n^*$ be the $n$-sided regular polygon. Assume that the polygons have the same fixed area, or the same fixed perimeter, and let $\lambda_1$ denote the first eigenvalue of the Dirac operator with infinite mass boundary conditions.

Regular polygon conjecture.

$$\lambda_1(P_n)\geq\lambda_1(P_n^*).$$

Thus the regular polygon is conjectured to minimise the first Dirac eigenvalue among $n$-gons under either area or perimeter constraint. This generalises the rectangle and triangle claims stated earlier in the paper; it is presented as a conjecture supported by the paper’s numerical shape-optimisation programme.